MOTION OF A PARTICLE 129 .Separating the variables and integrating we have Now suppose the body starts to fall from a distance r' from the center, then v = 0 when r = r' and c = — ®~-m Therefore r (6) gives the velocity at a distance r from the center. When the body falls from an infinite distance rf = oo and the velocity at any distance is (7) Therefore the velocity with which it will reach the surface •of the earth is miles (8) ~~ sec. If the body starts to fall from a distance above the surface equal to the radius of the earth, then in equation (6) r' = 2 r. Therefore _ v = Va . sec. Therefore about seventy-one per cent of the velocity attained in falling from an infinite distance is developed in the last 4000 miles. PROBLEMS. 1. A meteorite falls to the earth. Supposing it to start from infinity find the time it takes to travel the last 4000 miles. 2. A particle is attracted towards a fixed point by a force which varies inversely as the cube of the distance of the particle from the fixed point. Find the time it will take the particle to fall to the point if it starts from a distance d.